How do you find the equation of exponential decay?

Dec 9, 2014

${N}_{t} = {N}_{0} {e}^{- \lambda t}$

Exponential decay and growth occurs widely in nature so I will use radioactive decay as an example.

When an atom decays it is a random, chance event. The number of atoms decaying per second depends only on the number of undecayed atoms N.

So we can write:

Rate of decay:

$\frac{- N}{t} \propto N$

We can replace the $\propto$ sign with an = sign and the constant $\lambda$. We can also use calculus notation:

$- \frac{\mathrm{dN}}{\mathrm{dt}} = \lambda N$

Rearranging gives:

$\frac{\mathrm{dN}}{N} = - \lambda \mathrm{dt}$

Integrating both sides:

$\int \frac{\mathrm{dN}}{N} = - \lambda \int \mathrm{dt}$

So

$\ln N = - \lambda t + c$

If we apply the limits of integration such that when $t = 0$ , $N = {N}_{0}$ and when $t = t , N = {N}_{t}$ we get:

$\ln {N}_{t} - \ln {N}_{0} = - \lambda t$

So

$\ln \left({N}_{t} / {N}_{0}\right) = - \lambda t$

So

$\left({N}_{t} / {N}_{0}\right) = {e}^{- \lambda t}$

So

${N}_{t} = {N}_{0} {e}^{- \lambda t}$